module GroupRings using Nemo import Nemo: Group, GroupElem, Ring, parent, elem_type, parent_type import Base: convert, show, hash, ==, +, -, *, //, /, length, norm, rationalize, deepcopy_internal, getindex, setindex! ############################################################################### # # GroupRings / GroupRingsElem # ############################################################################### type GroupRing <: Ring group::Group pm::Array{Int,2} basis::Vector{GroupElem} basis_dict::Dict{GroupElem, Int} function GroupRing(G::Group; full=false) A = new(G) if full complete(A) end return A end end abstract AbstractGroupRingElem type GroupRingElem{T<:Number} <: AbstractGroupRingElem coeffs::AbstractVector{T} parent::GroupRing end export GroupRing, GroupRingElem ############################################################################### # # Type and parent object methods # ############################################################################### elem_type(::GroupRing) = GroupRingElem parent_type{T}(::GroupRingElem{T}) = GroupRing parent{T}(g::GroupRingElem{T}) = g.parent ############################################################################### # # GroupRing / GroupRingElem constructors # ############################################################################### function GroupRingElem{T<:Number}(c::AbstractVector{T}, RG::GroupRing) isdefined(RG, :basis) || complete(RG) length(c) == length(RG.basis) || throw("Can't create GroupRingElem -- lengths differ: length(c) = $(length(c)) != $(length(RG.basis)) = length(RG.basis)") GroupRingElem{T}(c,RG) end convert{T<:Number}(::Type{T}, X::GroupRingElem) = GroupRingElem(convert(AbstractVector{T}, X.coeffs), parent(X)) function (RG::GroupRing)(g::GroupElem, T::Type=Int) typeof(g) == elem_type(RG.group) || throw("$g does not belong to $(RG.group), the underlying group of $RG") g = try RG.group(g) catch throw("Can't coerce $g to the underlying group of $RG") end c = spzeros(T, length(RG.basis)) c[RG.basis_dict[g]] = one(T) return GroupRingElem(c, RG) end function GroupRing(G::Group, pm::Array{Int,2}) size(pm,1) == size(pm,2) || throw("pm must be of size (n,n), got $(size(pm))") return GroupRing(G, pm) end function GroupRing(G::Group, pm::Array{Int,2}, basis::Vector) size(pm,1) == size(pm,2) || throw("pm must be of size (n,n), got $(size(pm))") eltype(basis) == elem_type(G) || throw("basis must consist of elements of $G") basis_dict = reverse_dict(basis) return GroupRing(Group, pm, basis, basis_dict) end ############################################################################### # # Parent object call overloads # ############################################################################### function (RG::GroupRing)(T::Type=Int) isdefined(RG, :basis) || throw("Complete the definition of GroupRing first") return GroupRingElem(spzeros(T,length(RG.basis)), RG) end function (A::GroupRing)(X::GroupRingElem) length(X) == length(A.basis) || throw("Can not coerce to $A: lengths differ") X.parent = A return X end function (A::GroupRing)(x::AbstractVector) length(x) == length(A.basis) || throw("Can not coerce to $A: lengths differ") return GroupRingElem(x, A) end ############################################################################### # # Basic manipulation # ############################################################################### function deepcopy_internal(X::GroupRingElem, dict::ObjectIdDict) return GroupRingElem(deepcopy(X.coeffs), parent(X)) end function hash(X::GroupRingElem, h::UInt) return hash(X.coeffs, hash(parent(X), h)) end function getindex(X::GroupRingElem, n::Int) return X.coeffs[n] end function getindex(X::GroupRingElem, g::GroupElem) return X.coeffs[parent(X).basis_dict[g]] end function setindex!(X::GroupRingElem, k, n::Int) X.coeffs[n] = k end function setindex!(X::GroupRingElem, k, g::GroupElem) RG = parent(X) typeof(g) == elem_type(RG.group) || throw("$g is not an element of $(RG.group)") g = (RG.group)(g) X.coeffs[RG.basis_dict[g]] = k end ############################################################################### # # String I/O # ############################################################################### function show(io::IO, A::GroupRing) print(io, "Group Ring of [$(A.group)]") end function show(io::IO, X::GroupRingElem) if X == parent(X)() print(io, "0*$((parent(X).group)())") else T = eltype(X.coeffs) elts = ("$(X.coeffs[i])*$(parent(X).basis[i])" for i in 1:length(X) if X.coeffs[i] != zero(T)) join(io, elts, " + ") end end ############################################################################### # # Comparison # ############################################################################### function (==)(X::GroupRingElem, Y::GroupRingElem) parent(X) == parent(Y) || return false if eltype(X.coeffs) != eltype(Y.coeffs) warn("Comparing elements with different coeffs Rings!") end X.coeffs == Y.coeffs || return false return true end function (==)(A::GroupRing, B::GroupRing) return A.group == B.group end ############################################################################### # # Scalar operators # ############################################################################### (-)(X::GroupRingElem) = GroupRingElem(-X.coeffs, parent(X)) (*){T<:Number}(a::T, X::GroupRingElem{T}) = GroupRingElem(a*X.coeffs, parent(X)) function scalar_multiplication{T<:Number, S<:Number}(a::T, X::GroupRingElem{S}) promote_type(T,S) == S || warn("Scalar and coeffs are in different rings! Promoting result to $(promote_type(T,S))") return GroupRingElem(a*X.coeffs, parent(X)) end (*){T<:Number}(a::T,X::GroupRingElem) = scalar_multiplication(a, X) (/){T<:Number}(a::T, X::GroupRingElem) = scalar_multiplication(1/a, X) (//){T<:Rational, S<:Rational}(X::GroupRingElem{T}, a::S) = GroupRingElem(X.coeffs//a, parent(X)) (//){T<:Rational, S<:Integer}(X::GroupRingElem{T}, a::S) = X//convert(T,a) ############################################################################### # # Binary operators # ############################################################################### function add{T<:Number}(X::GroupRingElem{T}, Y::GroupRingElem{T}) parent(X) == parent(Y) || throw(ArgumentError( "Elements don't seem to belong to the same Group Ring!")) return GroupRingElem(X.coeffs+Y.coeffs, parent(X)) end function add{T<:Number, S<:Number}(X::GroupRingElem{T}, Y::GroupRingElem{S}) parent(X) == parent(Y) || throw(ArgumentError( "Elements don't seem to belong to the same Group Ring!")) warn("Adding elements with different base rings!") return GroupRingElem(+(promote(X.coeffs, Y.coeffs)...), parent(X)) end (+)(X::GroupRingElem, Y::GroupRingElem) = add(X,Y) (-)(X::GroupRingElem, Y::GroupRingElem) = add(X,-Y) function groupring_mult!(X,Y,pm,result) for (j,y) in enumerate(Y) if y != zero(eltype(Y)) for (i, index) in enumerate(pm[:,j]) if X[i] != zero(eltype(X)) index == 0 && throw(ArgumentError("The product don't seem to belong to the span of basis!")) result[index] += X[i]*y end end end end return result end function groupring_mult{T<:Number}(X::AbstractVector{T}, Y::AbstractVector{T}, pm::Array{Int,2}) result = zeros(X) return groupring_mult!(X,Y,pm,result) end function groupring_mult(X::AbstractVector, Y::AbstractVector, pm::Array{Int,2}) T = promote_type(eltype(X), eltype(Y)) result = zeros(T, deepcopy(X)) groupring_mult!(X, Y, pm, result) return result end function groupring_mult{T<:Number}(X::GroupRingElem{T}, Y::GroupRingElem{T}) parent(X) == parent(Y) || throw(ArgumentError( "Elements don't seem to belong to the same Group Ring!")) result = groupring_mult(X.coeffs, Y.coeffs, parent(X).pm) return GroupRingElem(result, parent(X)) end function groupring_mult{T<:Number, S<:Number}(X::GroupRingElem{T}, Y::GroupRingElem{S}) parent(X) == parent(Y) || throw("Elements don't seem to belong to the same Group Ring!") warn("Multiplying elements with different base rings!") result = groupring_mult(promote(X.coeffs,Y.coeffs)..., parent(X).pm) return GroupRingElem(result, parent(X)) end (*)(X::GroupRingElem, Y::GroupRingElem) = groupring_mult(X,Y) ############################################################################### # # Misc # ############################################################################### length(X::GroupRingElem) = length(X.coeffs) norm(X::GroupRingElem, p=2) = norm(X.coeffs, p) augmentation(X::GroupRingElem) = sum(X.coeffs) function rationalize{T<:Integer, S<:Number}(::Type{T}, X::GroupRingElem{S}; tol=eps(S)) v = rationalize(T, X.coeffs, tol=tol) return GroupRingElem(v, parent(X)) end function reverse_dict(a::AbstractVector) return Dict{eltype(a), Int}(x => i for (i,x) in enumerate(a)) end function create_pm{T<:GroupElem}(basis::Vector{T}, basis_dict::Dict{T, Int}, limit=length(basis); twisted=false) product_matrix = zeros(Int, (limit,limit)) for i in 1:limit x = basis[i] if twisted x = inv(x) end for j in 1:limit w = x*(basis[j]) product_matrix[i,j] = basis_dict[w] end end return product_matrix end function complete(A::GroupRing) if !isdefined(A, :basis) A.basis = collect(elements(A.group)) end if !isdefined(A, :basis_dict) A.basis_dict = reverse_dict(A.basis) end if !isdefined(A, :pm) A.pm = try create_pm(A.basis, A.basis_dict) catch err isa(err, KeyError) && throw("Product is not supported on basis") throw(err) end end return A end end # of module GroupRings